Refinement of regula falsi method for solving non-linear equations

Abstract

Searching for roots of non-linear equations in the real domain is a common problem in science and engineering. Even though a few closed-form analytical solutions exist for algebraic equations, there is no such formula for transcendental equations, which brings numerical approximate methods to the frontline. The classical Regula Falsi method (M₁) is a root-finding numerical technique with linear convergence and requires checking the bracket condition at each iteration. Recently, a new root-finding technique (M₂) is proposed, that is superlinearly convergent with order p = √2, and it does not require checking the bracket condition. The iterations of this method are advanced by the formula : x(n)₊₁ = λN(n)₋₁ + (1 − λ)N(n), 0 ≤ λ ≤ 1 for n = 1, 2, 3, ..., where N(m) = x(m) − f(x(m)) ′ ⁄ (x(m)), m = 0, 1, 2, ..., where f′(x) is the derivative of the function defined in the non-linear equation f(x) = 0. In fact, due to the parameter λ, M₂ generates a family of root-finding techniques, all of which are superlinearly convergent. The convergence analysis of this method has also been established. However, this method fails in some cases, such as when 1) Nonlinear functions are not sufficiently smooth, 2) Nonlinear functions are not explicitly known, and 3) Derivatives have complex formulas. To alleviate those issues, each term N(n)₋₁ and N(n) is replaced by the difference quotient D(n)₋₁ and D(n), respectively, to obtain the following method: (M₃) x(n)₊₁ = λD(n)₋₁+ (1 − λ)D(n), for 0 ≤ λ ≤ 1 for n = 1, 2, 3, ..., where D(m) = x(m) − [f(x(m))(f(x(m)) − f(x(m)₋₁))]/(x(m) − x(m)₋₁) for all m = 1, 2, 3, .... The accuracy and convergence efficiency of the proposed method M₃ are demonstrated by several numerical test examples. While M₃ achieves a better accuracy comparable to both M₁ and M₂, it converges to exact roots faster than both M₁ and M₂. For example, considering the initial approximations and x₀ = −0.4 , x₁ = 0.5 for the exact root r = 0 of the equation eˣsin x + ln(x² + 1) = 0, M₁ requires 51 iterations to give an approximation of 0.000000000000009, M₂ achieves of an approximation 0.0000000000000000 within 16 iterations, while M₃ reaches an approximation of 0.0000000000014123 in 12 iterations.